Given a deep polynomial neural network \(\mathcal {N}: \mathbb {R}^{d} \rightarrow \mathbb {R}^{c}\)  with depth \(h\)  and polynomial degree \(r\) , let \(m = \lbrace m_0, m_1,\ldots , m_h\rbrace \)  represent the architecture of \(\mathcal {N}\)  with the width \(m_i\)  of layer \(i\)  and \(m_0=d\) , \(m_h=c\) . Let \(\mathcal {F}_{m,r}\)  be the functional space of \(\mathcal {N}\) .
Kileel et al. [1]} define the functional variety \(\mathcal {V}_{m,r}\)  as \(\mathcal {V}_{m,r} = \overline{\mathcal {F}_{m,r}}\) , which is the Zariski closure of the functional space \(\mathcal {F}_{m,r}\) . “The Zariski closure of a set \(X\)  is the smallest set containing \(X\)  that can be described by polynomial equations.” [1]} The functional variaty \(\mathcal {V}_{m,r}\)  can be much larger than the functional space \(\mathcal {F}_{m,r}\) .
Kileel et al. [1]} propose to use the dimensionality of functional variety, denoted by \(\dim \mathcal {V}_{m,r}\) , as the representation of the expressive capacity of deep polynomial neural networks, written as
\(\nonumber MEC(\mathcal {N}_{m, r}) = \dim \mathcal {V}_{m,r}.\) 
